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10.5 Long-Memory Models 361
and
Cov(Y (t + h), Y (t)) b ′ e Ah b.
Inference for continuous-time ARMA processes is more complicated than for
continuous-time AR(1) processes because higher-order processes are not Markovian,
so the simple calculation that led to (10.4.6) must be modified. However, the
likelihood of observations at times t1,...,tn can still easily be computed using the
discrete-time Kalman recursions (see Jones, 1980).
Continuous-time ARMA processes with thresholds constitute a useful class of
nonlinear time series models. For example, the continuous-time threshold AR(1)
process with threshold at r is defined as a solution of the stochastic differential
equations
and
10.5 Long-Memory Models
dX(t) + a1X(t)dt b1 dt + σ1dB(t), X(t) ≤ r,
dX(t) + a2X(t)dt b2 dt + σ2dB(t), X(t) > r.
For a detailed discussion of such processes, see Stramer, Brockwell, and Tweedie
(1996). Continuous-time threshold ARMA processes are discussed in Brockwell
(1994) and non-Gaussian CARMA(p, q) processes in Brockwell (2001). For more
on continuous-time models see Bergstrom (1990) and Harvey (1990).
The autocorrelation function ρ(·) of an ARMA process at lag h converges rapidly to
zero as h →∞in the sense that there exists r>1 such that
r h ρ(h) → 0 as h →∞. (10.5.1)
Stationary processes with much more slowly decreasing autocorrelation function,
known as fractionally integrated ARMA processes, or more precisely as ARIMA
(p, d, q) processes with 0 < |d| < 0.5, satisfy difference equations of the form
(1 − B) d φ(B)Xt θ(B)Zt, (10.5.2)
where φ(z) and θ(z) are polynomials of degrees p and q, respectively, satisfying
φ(z) 0 and θ(z) 0 for all z such that |z| ≤1,
B is the backward shift operator, and {Zt} is a white noise sequence with mean 0 and
variance σ 2 . The operator (1 − B) d is defined by the binomial expansion
(1 − B) d ∞
πjB j ,
j0